Optimal. Leaf size=111 \[ \frac {2 i F\left (\left .\frac {1}{2} \left (i a-\frac {\pi }{2}+i b \log \left (c x^n\right )\right )\right |2\right ) \sqrt {i \sinh \left (a+b \log \left (c x^n\right )\right )}}{3 b n \sqrt {\sinh \left (a+b \log \left (c x^n\right )\right )}}+\frac {2 \cosh \left (a+b \log \left (c x^n\right )\right ) \sqrt {\sinh \left (a+b \log \left (c x^n\right )\right )}}{3 b n} \]
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Rubi [A]
time = 0.04, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2715, 2721,
2720} \begin {gather*} \frac {2 \sqrt {\sinh \left (a+b \log \left (c x^n\right )\right )} \cosh \left (a+b \log \left (c x^n\right )\right )}{3 b n}+\frac {2 i \sqrt {i \sinh \left (a+b \log \left (c x^n\right )\right )} F\left (\left .\frac {1}{2} \left (i a+i b \log \left (c x^n\right )-\frac {\pi }{2}\right )\right |2\right )}{3 b n \sqrt {\sinh \left (a+b \log \left (c x^n\right )\right )}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2715
Rule 2720
Rule 2721
Rubi steps
\begin {align*} \int \frac {\sinh ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac {\text {Subst}\left (\int \sinh ^{\frac {3}{2}}(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac {2 \cosh \left (a+b \log \left (c x^n\right )\right ) \sqrt {\sinh \left (a+b \log \left (c x^n\right )\right )}}{3 b n}-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {\sinh (a+b x)}} \, dx,x,\log \left (c x^n\right )\right )}{3 n}\\ &=\frac {2 \cosh \left (a+b \log \left (c x^n\right )\right ) \sqrt {\sinh \left (a+b \log \left (c x^n\right )\right )}}{3 b n}-\frac {\sqrt {i \sinh \left (a+b \log \left (c x^n\right )\right )} \text {Subst}\left (\int \frac {1}{\sqrt {i \sinh (a+b x)}} \, dx,x,\log \left (c x^n\right )\right )}{3 n \sqrt {\sinh \left (a+b \log \left (c x^n\right )\right )}}\\ &=\frac {2 i F\left (\left .\frac {1}{2} \left (i a-\frac {\pi }{2}+i b \log \left (c x^n\right )\right )\right |2\right ) \sqrt {i \sinh \left (a+b \log \left (c x^n\right )\right )}}{3 b n \sqrt {\sinh \left (a+b \log \left (c x^n\right )\right )}}+\frac {2 \cosh \left (a+b \log \left (c x^n\right )\right ) \sqrt {\sinh \left (a+b \log \left (c x^n\right )\right )}}{3 b n}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 0.10, size = 114, normalized size = 1.03 \begin {gather*} \frac {-2 \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};\cosh \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+\sinh \left (2 \left (a+b \log \left (c x^n\right )\right )\right )\right ) \sqrt {1-\cosh \left (2 \left (a+b \log \left (c x^n\right )\right )\right )-\sinh \left (2 \left (a+b \log \left (c x^n\right )\right )\right )}+\sinh \left (2 \left (a+b \log \left (c x^n\right )\right )\right )}{3 b n \sqrt {\sinh \left (a+b \log \left (c x^n\right )\right )}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 7.13, size = 143, normalized size = 1.29
method | result | size |
derivativedivides | \(\frac {-\frac {i \sqrt {1-i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \sqrt {2}\, \sqrt {1+i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \sqrt {i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \EllipticF \left (\sqrt {1-i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}, \frac {\sqrt {2}}{2}\right )}{3}+\frac {2 \left (\cosh ^{2}\left (a +b \ln \left (c \,x^{n}\right )\right )\right ) \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}{3}}{n \cosh \left (a +b \ln \left (c \,x^{n}\right )\right ) \sqrt {\sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, b}\) | \(143\) |
default | \(\frac {-\frac {i \sqrt {1-i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \sqrt {2}\, \sqrt {1+i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \sqrt {i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \EllipticF \left (\sqrt {1-i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}, \frac {\sqrt {2}}{2}\right )}{3}+\frac {2 \left (\cosh ^{2}\left (a +b \ln \left (c \,x^{n}\right )\right )\right ) \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}{3}}{n \cosh \left (a +b \ln \left (c \,x^{n}\right )\right ) \sqrt {\sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, b}\) | \(143\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.09, size = 171, normalized size = 1.54 \begin {gather*} -\frac {2 \, {\left (\sqrt {2} \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \sqrt {2} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )} {\rm weierstrassPInverse}\left (4, 0, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right ) - {\left (\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 2 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 1\right )} \sqrt {\sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}}{3 \, {\left (b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + b n \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sinh ^{\frac {3}{2}}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\mathrm {sinh}\left (a+b\,\ln \left (c\,x^n\right )\right )}^{3/2}}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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